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Wikipedia / Linear time-invariant system

In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined in the overview below. These properties apply (exactly or approximately) to many important physical systems, in which case the response of the system to an arbitrary input can be found directly using convolution: where is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication). What's more, there are systematic methods for solving any such system (determining ), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.

Linear time-invariant system theory is also used in image processing, where the systems have spatial dimensions instead of, or in addition to, a temporal dimension. These systems may be referred to as linear translation-invariant to give the terminology the most general reach. In the case of generic discrete-time (i.e., sampled) systems, linear shift-invariant is the corresponding term. LTI system theory is an area of applied mathematics which has direct applications in electrical circuit analysis and design, signal processing and filter design, control theory, mechanical engineering, image processing, the design of measuring instruments of many sorts, NMR spectroscopy, and many other technical areas where systems of ordinary differential equations present themselves.

Overview

The defining properties of any LTI system are linearity and time invariance.

  • Linearity, . The latter condition is often referred to as the superposition principle.

  • Time invariance means that whether we apply an input to the system now or T seconds from now, the output will be identical except for a time delay of T. Hence, the system is time invariant because the output does not depend on the particular time the input is applied.

Through these properties, it is reasoned that LTI systems can be characterized entirely by a single function called the system's impulse response, as, by superposition, any arbitrary signal can be expressed as a superposition of time-shifted impulses is simply the convolution. This is called a continuous time system. Similarly, a discrete-time linear time-invariant (or, more generally, "shift-invariant") system is defined as one operating in discrete time where y, x, and h are sequences and the convolution, in discrete time, uses a discrete summation rather than an integral.

LTI systems can also be characterized in the frequency domain by the system's transfer function, which for a continuous-time or discrete-time system is the Laplace transform or Z-transform of the system's impulse response, respectively. As a result of the properties of these transforms, the output of the system in the frequency domain is the product of the transfer function and the corresponding frequency-domain representation of the input. In other words, convolution in the time domain is equivalent to multiplication in the frequency domain.

For all LTI systems, the eigenfunctions, and the basis functions of the transforms, are complex exponentials. The output signal will be shifted in phase and amplitude, but always with the same frequency upon reaching steady-state. LTI systems cannot produce frequency components that are not in the input.

LTI system theory is good at describing many important systems. Most LTI systems are considered "easy" to analyze, at least compared to the time-varying and/or nonlinear case. Any system that can be modeled as a linear differential equation with constant coefficients is an LTI system. Examples of such systems are electrical circuits made up of resistors, inductors, and capacitors (RLC circuits). Ideal spring–mass–damper systems are also LTI systems and are mathematically equivalent to RLC circuits.

Most LTI system concepts are similar between the continuous-time and discrete-time cases. In image processing, the time variable is replaced with two space variables, and the notion of time invariance is replaced by two-dimensional shift invariance. When analyzing filter banks and MIMO systems, it is often useful to consider vectors of signals. A linear system that is not time-invariant can be solved using other approaches such as the Green function method.

Continuous-time systems

Impulse response and convolution

The behavior of a linear, continuous-time, time-invariant system with input signal x(t) and output signal y(t) is described by the convolution integral:

  1. {|

(using commutativity) |}

is the system's response to an impulse, and the system is said to be causal.

, the output function is just constant, and the system is uninteresting.

must satisfy :

And the time-invariance requirement is:

In this notation, we can write the impulse response

Similarly:

  1. {|

(using ) |}

which has the form of the right side of

, can be represented by a continuum of time-shifted impulse functions, combined "linearly", as shown at . The system's linearity property allows the system's response to be represented by the corresponding continuum of impulse responses, combined in the same way. And the time-invariance property allows that combination to be represented by the convolution integral.

The mathematical operations above have a simple graphical simulation.

Exponentials as eigenfunctions

An eigenfunction where f is the eigenvalue, a constant.

The exponential functions, are eigenfunctions of a linear, time-invariant which, by the commutative property of convolution

is dependent only on the parameter s.

is an eigenfunction of an LTI system, and the corresponding eigenvalue.

Direct proof

It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems.

a time-shifted version of it.

.

.

as input will give a complex exponential of same frequency as output.

Fourier and Laplace transforms

The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The one-sided Laplace transform). The Fourier transform are called the system function, system response, or transfer function.

The Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of t less than some value. Usually, this "start time" is set to zero, for convenience and without loss of generality, with the transform integral being taken from zero to infinity (the transform shown above with lower limit of integration of negative infinity is formally known as the bilateral Laplace transform).

The Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it cannot be directly applied to input and output signals that are not square integrable. The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite signals via the Wiener–Khinchin theorem even when Fourier transforms of the signals do not exist.

One can use the system response directly to determine how any particular frequency component is handled by a system with that Laplace transform. If we evaluate the system response (Laplace transform of the impulse response) at complex frequency , where , we obtain |H(s)| which is the system gain for frequency f. The relative phase shift between the output and input for that frequency component is likewise given by arg(H(s)).

Examples

Important system properties

Some of the most important properties of a system are causality and stability. Causality is a necessity for a physical system whose independent variable is time, however this restriction is not present in other cases such as image processing.

Causality

is the impulse response. It is not possible in general to determine causality from the two-sided Laplace transform. However, when working in the time domain, one normally uses the one-sided Laplace transform which requires causality.

Stability

A system is bounded-input, bounded-output stable

(that is, a finite maximum absolute value, the impulse response, is in L1 (has a finite L1

In the frequency domain, the region of convergence.

As an example, the ideal low-pass filter with impulse response equal to a sinc function is not BIBO stable, because the sinc function does not have a finite L1 and equal to a sinusoid at the cut-off frequency, then the output will be unbounded for all times other than the zero crossings.

Discrete-time systems

Almost everything in continuous-time systems has a counterpart in discrete-time systems.

Discrete-time systems from continuous-time systems

In many contexts, a discrete time (DT) system is really part of a larger continuous time (CT) system. For example, a digital recording system takes an analog sound, digitizes it, possibly processes the digital signals, and plays back an analog sound for people to listen to.

is a CT signal, then the sampling circuit used before an analog-to-digital converter where T is the sampling period. Before sampling, the input signal is normally run through a so-called Nyquist filter which removes frequencies above the "folding frequency" 1/(2T); this guarantees that no information in the filtered signal will be lost. Without filtering, any frequency component above the folding frequency (or Nyquist frequency) is aliased to a different frequency (thus distorting the original signal), since a DT signal can only support frequency components lower than the folding frequency.

Impulse response and convolution

A discrete system

Note that unless the transform itself changes with n, the output sequence is just constant, and the system is uninteresting. (Thus the subscript, n.) In a typical system, y[n] depends most heavily on the elements of x whose indices are near n.

For the special case of the Kronecker delta function the output sequence is the impulse response

must satisfy:

And the time-invariance requirement is:

in terms of a sum of weighted delta functions.

where we have invoked .

And because of

Therefore:

  1. {|

(commutativity) |}

can therefore be interpreted as proportional to a weighted average of the function x[k]. The weighting function is h[−k], simply shifted by amount n. As n changes, the weighting function emphasizes different parts of the input function. Equivalently, the system's response to an impulse at n=0 is a "time" reversed copy of the unshifted weighting function. When h[k] is zero for all negative k, the system is said to be causal.

Exponentials as eigenfunctions

An eigenfunction

where f is the eigenvalue, a constant.

The exponential functions, are eigenfunctions of a linear, time-invariant. A simple proof illustrates this concept.

which is equivalent to the following by the commutative property of convolution is dependent only on the parameter z.

is an eigenfunction.

Z and discrete-time Fourier transforms

The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Z transform

is exactly the way to get the eigenvalues from the impulse response.. The discrete-time Fourier transform gives the eigenvalues of pure sinusoids are called the system function, system response, or transfer function.

Like the one-sided Laplace transform, the Z transform is usually used in the context of one-sided signals, i.e. signals that are zero for t<0. The discrete-time Fourier transform Fourier series may be used for analyzing periodic signals.

Just as with the Laplace transform transfer function in continuous-time system analysis, the Z transform makes it easier to analyze systems and gain insight into their behavior.

Examples

Important system properties

. Two of the most important properties of a system are causality and stability. Non-causal (in time) systems can be defined and analyzed as above, but cannot be realized in real-time. Unstable systems can also be analyzed and built, but are only useful as part of a larger system whose overall transfer function is stable.

Causality

A discrete-time LTI system is causal if the current value of the output depends on only the current value and past values of the input. is the impulse response. It is not possible in general to determine causality from the Z transform, because the inverse transform is not unique. When a region of convergence is specified, then causality can be determined.

Stability

A system is bounded input, bounded output stable

(that is, if bounded input implies bounded output, in the sense that the maximum absolute values

In the frequency domain, the region of convergence must contain the unit circle (i.e., the locus for complex z).

Notes

See also

References

Further reading